We now aim to show that the right-hand vertical map is a weak equivalence.
To that end, we consider the diagram \(\mathrm {D}\underline{U}_{(-)} \colon sA^\mathrm {op}\rightarrow {\Delta {T}\mathrm {op}}\) .
For a topological space \(T\) , let \(\mathcal {O}(T)\)  denote its partially ordered set of open subsets.
Then, we can write \(\mathrm {D}\underline{U}_{(-)}\)  as a diagram \(\mathrm {D}\underline{U}_{(-)} \colon sA^\mathrm {op}\rightarrow \mathcal {O}(\mathrm {D}M)\) .
For \(x \in \mathrm {D}\underline{M}\) , let \((sA^\mathrm {op})_x \subset sA^\mathrm {op}\)  denote the full subcategory of \(sA^\mathrm {op}\)  on those objects \(\alpha \in sA\)  satisfying that \(x \in U_\alpha \subset M\) .
Since \({{U}}\)  is an open covering of \(M\) , the category \((sA^\mathrm {op})_x\)  is non-empty and filtered, for each \(x \in M\) .
We can therefore apply Lurie's Seifert-van Kampen Theorem [1]} (which, in this particular incarnation is often called the Nerve Theorem):
the map
\({\mathrm {hocolim}}^{{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\big ( sA^\mathrm {op}\xrightarrow{} {\Delta {T}\mathrm {op}}\xrightarrow{} {{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\big )\longrightarrow \operatorname{Sing}\mathrm {D}M\) 
